Generalized Linear Models

**Lecture Times**: Tuesday, Thursday 10:00- 11:45am, BE 358

**Instructor**: Tatiana Xifara

**Email**: xifara@soe.ucsc.edu

**Office**: Baskin Engineering 365B

**Office Hours:** Monday 2:00-3:30pm or by appointment.

**Required Text:** The course will not use a textbook. Good reference textbooks include "Generalised Linear Models" by McCullagh & Nelder, "Categorical Data Analysis" by Agresti and "Generalised Linear Models: A Bayesian Perspective" by Dey, Chosh and Mallick. An extended list of references will be given in class.

**Course Objectives:** This is a graduate-level course on basic theory, methodology and applications of Generalized Linear Models (GLMs). Emphasis will be placed on statistical modeling, building from standard normal linear models, extending to GLMs, and going beyond GLMs. The course will cover both frequentist inference and Bayesian methodology approaches but the focus will be Bayesian methodology. The course will contain an introductory overview of Bayesian methods for analysis of binomial, count, categorical, and event time data. In particular, within the Bayesian modeling framework, we will discuss particularly important hierarchical extensions of the standard GLM setting. We will be using the statistical software R to illustrate the methods with examples and case studies.

**Tentative Schedule: **We will cover topics from the following:

**1. Introduction to GLMs**

- Statistical modeling in the context of GLMs and motivating examples
- Exponential family of distributions
- Components of a GLM, examples of GLMs

**2. Likelihood inference for GLMs**

- Likelihood estimation (iterative weighted least squares), existence and uniqueness of MLEs
- Model fitting, model diagnostics (residuals for GLMs), analysis of deviance
- model and variable selection, Goodness-of-fit in GLMs

**3. Bayesian inference in GLMs (basics)**

- General setting, examples, priors for GLMs
- Comparison with frequentist approach, posterior computations
- MCMC strategies (
*Gibbs, Metropolis-Hastings*) - Bayesian Residuals analysis and model choice

**4. Regression models for categorical responses and count data**

- Models for binary responses (dose-response modeling, logistic regression and generalizations)
- Count data (poisson regression and poisson log-linear models)
- Data augmentation algorithms

**5. Bayesian inference in GLMs (continued)**

- Hierarchical GLMs, overdispered GLMs
- Generalized linear mixed models
- Longitudinal data analysis